Catalan triangle numbers and binomial coefficients

TL;DR

This paper explores the relationships between binomial coefficients, Catalan triangle numbers, and their connections to Lie algebras, introducing the alternating Jacobsthal triangle and q-deformations with combinatorial significance.

Contribution

It establishes a novel representation of binomial coefficients as sums along Catalan triangle rows and introduces the alternating Jacobsthal triangle with its combinatorial properties and q-deformations.

Findings

Binomial coefficients can be expressed as weighted sums along Catalan triangle rows.

The alternating Jacobsthal triangle exhibits interesting combinatorial subsequences.

q-deformations of these sequences reveal new combinatorial structures.

Abstract

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions. The generating functions of these sequences enable us to define their k-analogue of q-deformation. We show that this deformation also gives rise to interesting combinatorial sequences. The starting point of this work is certain identities in the study of Khovanov--Lauda--Rouquier algebras and fully commutative elements of a Coxeter group.